Sequential

Sequential colormaps are good for representing magnitude of data.
How much flow, how much precipitation, how much weight, temperature, etc.
Having a monotonic
lightness factor
is important for perceptual consistency.

Non-linear lightness is used to emphasize certain ranges of data, perhaps where snow changes to ice or rain.
Non-monotonic lightness can be used to emphasize different types of precipitation or phase changes, etc.

Reversed colormap

Reversed colormaps are useful for sparse data such as astronomical images or precipitation data where a lot of data is at or near zero relative to other data.
The reversal leads to near-zero areas being white and higher intensities being darker.
While any colormap can be reversed, typically sequential colormaps are used with/without reversal.

Matplotlib

Matplotlib colormaps are reversed by appending _r to the colormap name.
For example:

cmap='cubehelix_r'

Matlab / Octave

Matlab and GNU Octave colormaps are reversed by flipud() the colormap.
Colormaps in .m code are represented as an (N,3) array, where N is the number of steps in the colormap (typically 64 or 256).

colormap(flipud(cubehelix()))

Diverging

Diverging colormaps are useful for positive or negative data where the sign is as important as the magnitude.
For example, in/out flows, positive/negative charge.
These colormaps are white near the zero point (which can be offset) and intensify as their absolute magnitude increases.

Qualitative

These colormaps emphasize difference between values, but without a particular sense of ordering.
This can be useful for categories, say a histogram of salary vs. employee type.